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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fquq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fquq.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/s_transition/fquq.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Properties with extended structural successor for closures ***************)
+
+lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+                             ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
+                             ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #K #V #K1 #H
+  elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
+  elim (lifts_total V (𝐔❴1❵)) #T #HVT
+  /3 width=5 by cpx_delta, fqu_drop, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #T #U #HTU #K1 #H
+  elim (lpx_inv_bind_dx … H) -H #I0 #K0 #HK0 #HI0 #H destruct
+  /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+                             ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
+                             ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #X #H
+  elim (lpx_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
+  /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
+  elim (lpx_inv_unit_sn … H) -H #K2 #HLK2 #H destruct
+  /3 width=5 by cpx_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
+| /3 width=5 by lpx_bind_refl_dx, fqu_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with extended optional structural successor for closures ******)
+
+lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+                              ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
+                              ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
+[ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+                              ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
+                              ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
+[ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.